Denominator vectors and compatibility degrees in cluster algebras of finite type

TL;DR

This paper provides two linear algebra-based descriptions of denominator vectors in finite type cluster algebras, offering simple proofs for their non-negativity and non-zero properties across all cluster variables.

Contribution

It introduces new descriptions of denominator vectors using compatibility degrees and root functions, simplifying proofs of their fundamental properties.

Findings

Denominator vectors are non-negative and non-zero for all cluster variables.

Two new descriptions of denominator vectors are provided, based on compatibility degrees and root functions.

The proofs rely solely on linear algebra, simplifying previous approaches.

Abstract

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the denominator vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.